abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Background
Basic concepts
equivalences in/of -categories
Universal constructions
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Theorems
Extra stuff, structure, properties
Models
A duality axiom used in various branches of synthetic mathematics, such as
The axiom of synthetic quasi-coherence is a more general version of the Kock-Lawvere axiom in synthetic differential geometry as it can apply to algebras of a generic model of a Horn theory rather than just the Weil algebras of a generic local ring.
Let be a Horn theory and let be the generic model of the theory . A homomorphism between two models of is a function which preserves all the signatures of the theory . An -algebra consists of a -model with a homomorphism . An -algebra homomorphism on an -algebra is a homomorphism from to . The spectrum of is the set of -algebra homomorphisms:
An -algebra is quasi-coherent if the canonical evaluation homomorphism
is an isomorphism.
The axiom SQCP or synthetic quasi-coherence for the univariate polynomial algebra states that the free -algebra on one generator is quasi-coherent.
Phoa's principle for distributive lattices is equivalent in strength to the axiom SQCP for -algebras where is the Lawvere theory of distributive lattices.
An -algebra is finitely quotiented if there is a natural number and two finite families of elements in the generic model such that is isomorphic to the quotient algebra .
The axiom SQCI or synthetic quasi-coherence for finitely quotiented algebras states that every finitely quotiented -algebra of is quasi-coherent.
In applications to synthetic algebraic geometry and synthetic (infinity,1)-category theory, one usually considers the notion of synthetic quasi-coherence as applying to finitely presented algebras: A finitely presented -algebra is an -algebra which is isomorphic to a quotient of the free -algebra on a finite set of generators.
The axiom SQCF or synthetic quasi-coherence for finitely presented algebras states that all finitely presented -algebras are quasi-coherent.
In applications to synthetic topology, such as synthetic Stone duality, as well as synthetic domain theory, one usually considers the notion of synthetic quasi-coherence as applying to countably presented algebras: A countably presented -algebra is an -algebra which is isomorphic to a quotient of the free -algebra on a countable set of generators.
The axiom SQCC or synthetic quasi-coherence for countably presented algebras states that all countably presented -algebras are quasi-coherent.
In applications to synthetic differential geometry, one considers Artinian -algebras, i.e. -algebras that satisfy the descending chain condition.
The axiom SQCA or synthetic quasi-coherence for Artinian algebras states that all Artinian -algebras are quasi-coherent.
The Kock-Lawvere axiom is the axiom of synthetic quasi-coherence for Artinian -algebras with the Horn theory of a local ring (hence by the definitions given on this page, every -algebra is a local ring and every Artinian -algebra is a Weil algebra).
Ingo Blechschmidt, Using the internal language of toposes in algebraic geometry, PhD thesis (2017) [pdf, pdf]
Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)
Felix Cherubini, Thierry Coquand, Matthias Hutzler: A foundation for synthetic algebraic geometry, Mathematical Structures in Computer Science 34 Special Issue 9: Advances in Homotopy type theory (2024) 1008-1053 [doi:10.1017/S0960129524000239, arXiv:2307.00073]
Felix Cherubini, Thierry Coquand, Freek Geerligs, Hugo Moeneclaey, A Foundation for Synthetic Stone Duality (arXiv:2412.03203)
Leoni Pugh, Jonathan Sterling, When is the partial map classifier a Sierpiński cone? (arXiv:2504.06789)
Jonathan Sterling, Lingyuan Ye, Domains and Classifying Topoi (arXiv:2505.13096)
Last revised on June 23, 2025 at 23:41:34. See the history of this page for a list of all contributions to it.